       What kind of math problem is this?

• To: mathgroup at smc.vnet.net
• Subject: [mg20896] What kind of math problem is this?
• From: "Seth Chandler" <SChandler at uh.edu>
• Date: Sat, 20 Nov 1999 01:07:09 -0500 (EST)
• Organization: University of Houston
• Sender: owner-wri-mathgroup at wolfram.com

```This may be a little off topic, but I was hoping someone here could help.

Let r and f be real numbers between 0 and 1 inclusive. How does one
determine if there is a function u that satisfies this relation and, if so,
what the function is.

InverseFunction[u][f u[x] + (1 - f)u[y]] == (1 - r)(f x + (1 - f)y) + r x

For those interested in the origins of this math problem, it is as follows.
The "certainty equivalent wealth" associated with some lottery in which the
payoffs are x (the bad payoff) and y (the good payoff) is traditionally
thought to be equal to the inverse utility of the expected utility of the
lottery. That is, it is thought to be the amount which, if held with
certainty, would provide the same "utility" as the lottery.  Sometimes, this
formulation of certainty equivalent wealth is a bit messy.

Thus, others have proposed a simpler formulation that does not rely on
utility functions. Under this method, which game theorist Martin Shubik and
others have used, the certainty equivalent wealth of a lottery is set equal
to something between the worst case of the lottery and the expected value of
the lottery. The "where" in between is determined by a coefficient of risk
aversion (r). If r is zero, the lottery is valued at the expected value. If
r is one, the lottery is valued at the worst case.

My question is whether there is any utility function that could emulate the
behavior of this simpler formulation. Unfortunately, I can't figure out what
branch of mathematics is relevant to the question, or, more importantly, I
supposed, how to solve it. If Mathematica could be used in finding the